I see that there is an express train from Rome's FCO airport to the Termini train station for 9.5 euro. Is this the cheapest option to get to a hotel in the city? Do any hotels (that are reasonably priced) offer free shuttle pick up? There will be two adults and a twelve year old in my party.
The train is now 11 euro per person. Unless your hotel is near Termini I would take a taxi from the airport. Taxis now have a fixed rate from the airport to the city center of 40 euro, including luggage. The door-to-door service is great with baggage. Be sure to verify the price before you get in the taxi. Take a taxi from the taxi stand, which is clearly marked at the exits.
I would take a cab with that many people. The only warning I would give if you haven't been in Rome before is that Roman drivers are, to be polite, quite insane although safe and you should be prepared for anything. As Rick says in his book "Roman drivers stay in their lanes about as well as rocks do in an avalanche."
I don't know of any free shuttle services but there are ones that will drop you off at your hotel. Just google rome+airport shuttle or similar and quite a few will come up. You can compare the prices with a cab and then decide.
We're going next month and are staying at a Hilton in downtown Rome. Hotel suggested a taxi to prevent hauling luggage through the streets. Anyone have any alternative thoughts? This is our first trip and there are three people.
Took Helen's suggestion and Googled Rome + Shuttle. Door to door shuttles are VERY expensive, but I found the following link which is entitled "HOW TO GET FROM LEONDARDO DA VINCI (ROME FIUMICINO) AIRPORT
AND CIAMPINO AIRPORT TO ROME CITY CENTRE AND BACK.
http://www.initaly.com/romeair.htm
I've used the initaly.com site before and like this one, it is very informative.